Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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34 views

Direct proof for $\mathcal{O}_{\mathbb{CP}_{1}}(-2) \cong T^{*}\mathbb{CP}_{1}$

Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the ...
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78 views

Polarization of abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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22 views

Does the canonical line bundle coincide with dualizing sheaf when the base scheme is arbitrary

Let $X$ be a smooth projective curve over an algebarically closed field $k$. Consider the commutative diagram where the left square is cartesian. where $S,S'$ are schemes over $k$ and $X_S:=X\times ...
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1answer
87 views

Remembering Riemann-Roch

Embarrassingly, I've always struggled to remember the form of the Riemann-Roch theorem for curves. Does anyone have any intuition to share about how to remember the some of the terms in the formula? ...
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61 views

An application of Serre Duality

This is problem 2 of Chapter 5 in Voisin's book Hodge Theory and Complex Algebraic Geometry I. Let $X$ be a connected compact complex manifold of dimension $n$ and let $L$ be a holomorphic line ...
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1answer
35 views

Line Bundle with Everywhere Nonzero Section is Trivial

I'd like to show that a topological line bundle $(\pi,E,B)$ is trivial if there exists a section $\sigma : B \to E$ such that $\sigma(b) \neq 0$ for all $b$. I've been considering the map $B \times \...
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1answer
33 views

Canonical bundle and canonical divisor in a $K3$ surface

The algebraic definition of a $K3$ surface is this: A smooth algebraic suface $X$ is called $K3$ if: i) $X$ has trivial canonical bundle; ii) $h^1(X,\mathcal{O}_X)=0$. I know that i) ...
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44 views

Sections of very ample line bundle

Let $f: C \to D$ a dominant morphism which is not an isomorphism between two irreducible, reduced, projective curves $C,D$ over an alg closed field $k$ (unsure if algebraically closedness is ...
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38 views

When is the image of a line bundle again a line bundle

Hello everybody Motivation of my question Let $X$ be a scheme. Given a morphism $\mathcal{L}\overset{\beta}\to\mathcal{O}_X$ of line bundles over $X$. I want to understand under what conditions the ...
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23 views

Extending meromorphic function of line bundles

Let $X$ be a smooth irreducible curve over a field $k$, let $S$ be any scheme over $k$ and let $x_1,\cdots ,x_n\in X(S)$. Denote by $X_S=S\times_{\mathrm{Spec}(k)}X$ the base change and let $U=X_S-\...
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1answer
63 views

Line bundles have flat connections

Let $M$ be a manifold of $L \to M$ a line bundle (say over $\mathbb{C}$, ie complex line bundle). Is it true & why that for every such line bundle there exist a flat connection $\nabla_L : \Gamma(...
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1answer
33 views

Geometry of taking the sheaf of algebras associated to a line bundle.

Given a line bundle $L$ on a scheme $X$, we can construct the sheaf of $O_X$ algebras $\bigoplus\limits_{n=0}^\infty L^n$, which by the global Spec functor induces a map from some other scheme to $X$. ...
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34 views

Are global sections of a line bundle canonically identified with rational functions while rational sections are not?

Every line bundle $L$ on a smooth complex projective variety $X$ is of the form $\mathcal O(D)$ for some divisor $D$ on $X$. The global sections of $\mathcal O(D)$ are identified with the set of ...
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31 views

Ratio of sections of a line bundle

Does the ratio of two global sections of a complex line bundle on a compact Riemann surface define a rational function on this Riemann surface? Is the degree of this rational function (as a map from ...
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29 views

Line bundles on projective space and disk

I'm having a difficult time solving some exercice. I should prove the following : Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic ...
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120 views

Tautological bundle: algebraic geometry vs topology

I'm going to compare the two construction of twisted sheaf/bundle $\mathcal{O}(1)$ from algebraic and topological viewpoint: 1) Algebraic construction (Hartshorne's Algebraic Geometry, p. 117): ...
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36 views

Is Wronskian a Line Bundle for Riemann surfaces?

Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...
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42 views

Spin structure on a torus

Definition: A spin structure on a Riemannian manifold $M$ is a complex vector bundle S→M together with a isomorphism of algebra bundles Cl(M)→End(S). Here Cl(M) denotes the Clifford bundle over the ...
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1answer
65 views

Does every real line bundle admit a flat connection?

Consider a three-fold intersection $U_{i} \cap U_{j} \cap U_{k}$ of a trivialising Leray cover $\{\mathcal{U}\}$ for a real line bundle L over a $C^k$ manifold $M$. We have $g_{ij}g_{jk}g_{ki} = 1$ ...
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1answer
81 views

Numerically Trivial Invertible Sheaf

Vakil's Algebraic Geometry gives the following definition of numerically trivial invertible sheaf: Suppose $X$ is a proper k-variety, and $\mathscr{L}$ is an invertible sheaf on $X .$ If $i: C \...
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1answer
105 views

Definition of very ample line bundle.

I am reading Vakil's algebraic geometry. He gives a definition of very ample line bundle as followings: Suppose $\pi: X \rightarrow \operatorname{Spec} A$ is a proper morphism, and $\mathscr{L}$ is ...
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84 views

What is the Chern class of a line bundle over a number ring?

Question: Let $F$ be a finite extension over $\def\q{\mathbb Q}\q$. Let $\mathcal O_F$ be the integral closure of $\mathbb Z$ in $F$. Then if I am not mistaken, a line bundle (an invertible sheaf) ...
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17 views

Finding a refinement $\mathcal{W}$ of two covers of $\mathbb{P}^1$

Let $U_0 = \{s \neq 0\}$ and let $U_1 = \{t\neq0\}.$ Then the collection $\mathcal{U} = \{U_0, U_1\}$ gives an open cover of $\mathbb{P}^1.$ Similarly, if $V_0 = \{s \neq t\}$ and $V_1 = \{s \neq -t\},...
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1answer
22 views

Image of $c_1: H^1(\mathbb CP^n, \mathcal O^*)\to H^2(\mathbb CP^n,\mathbb Z)$

By the exact sequence $0\to \mathbb Z\to \mathcal O\to \mathcal O^*\to0$ and $H^1(\mathbb CP^n, \mathcal O)=0$ we can get the injection $$\text {Pic}(\mathbb CP^n)\cong H^1(\mathbb CP^n, \mathcal O^*)...
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1answer
28 views

The corresponding $\mathcal O(d)$ of the hyper surface of degree $d$

Let $X\subset\mathbb CP^n$ be the complex hypersurface cut out as $F = 0$ for a homogeneous polynomial $F$ of degree $d$. $X$ as a divisor, defines a line bundle $\mathcal O_{\mathbb CP^n}(n)$ for ...
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85 views

On picard group of an irreducible curve inside a smooth hypersurface

Let, $S$ be a smooth hypersurface in $\mathbb{P}^{3}$ and let $C$ be a reducible curve inside $S$ and $C =C_1 \cup C_2 \cup ...\cup C_n$, be its decomposition in terms of irreducible components. At ...
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3answers
167 views

Is compliment of a zero section of a vector bundle a submanifold?

Let $\pi:E\rightarrow M$ be a smooth vector bundle. Let $S:M\rightarrow E$ be it's zero section. Let $M'=E-S(M)$. Is $M'$ a smooth submanifold of $M$ ? It is clear that $S$ is a smooth injective ...
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1answer
47 views

On Isomorphism of stalk of a line bundle

Let, $X$ be a scheme and $L$ be a line bundle on $X$,then we know that at any point $x \in X$ we have $L_x \cong \mathcal O_x$. At this point my question is the following : Under this isomorphism is ...
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90 views

Pushforward of a globally generated line bundle

Let,$S$ be a smooth projective surface and $C \subset S$ be a smooth irreducible curve over an algebraically closed field $\mathbb K$ of characteristic $0$.Let, $j:C \to S$ be the inclusion morphism. ...
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60 views

Seesaw theorem for the pullback of a line bundle via projection map.

I am trying to understand Mumford's statement of the Seesaw theorem given at the start of Chapter 3 in his book Abelian Varieties. The statement is: Let $X$ be a complete variety, $Y$ a scheme (both ...
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69 views

Global sections of a line bundle

let $X$ be a normal projective curve over field $k$. let $Z=\sum_{x \text{ closed}} n_x [x]$ be a $0$-cycle on $X$ and $D \in Div(X)$ a Cartier divisor such that $[D]=Z$. then $D$ gives rise for a ...
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51 views

dual of the global section of line bundle $\mathscr{O}(1)$

I'm reading about the line bundles on projective spaces and there is something that I couldn't make sense of. Maybe I'm missing some definition so please correct me. Over $\mathbb{P}_k^1$, I know ...
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21 views

What is the relation between structure groups or transition functions of two isomorphic vector bundles?

Vector bundle $S=\Lambda^{\bullet}T^*M \otimes |\det TM|^{\frac{1}{2}}$ is called an spinor bundle for the bundle $TM\oplus T^*M$. it is an associated bundle to a $Spin(n,n)$-principal bundle where ...
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73 views

Blow up of a Kähler manifold is Kähler

Let $X$ be a compact Kähler manifold and $x\in X$. Denoty $Bl_x X$ the blow-up of $X$ at $x$ and $E$ the exceptional divisor. I want to see why $Bl_x X$ is Kähler. I know the idea is to take the ...
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91 views

Picard group of curves

I suppose the Picard group of a complete curve $C$ in $\mathbb P^n$ of degree $d>2$ is complicated. So, if we remove a general point $x\in C$ and denote $C'=C-x$, I think we have $\mathcal O_{C'}(1)...
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41 views

Distributions with values in line bundles

Let $M$ be a compact complex manifold, and let $L \rightarrow M$ be a holomorphic line bundle. I'm confused by the notion of a distribution that takes values in $L^{-1}$, and I'm looking for ...
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1answer
41 views

Any nonzero meromorphic $1$-form on a compact Riemann surface has degree $2g-2$

I am reading "Compact Riemann Surfaces" by Raghavan Narashimhan. Say X be a compact Riemann surface; after proving that the degree of the canonical bundle $K_X$ is $2g-2$ (using Riemann-Roch), where $...
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51 views

Is $\mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}$ a non-ample line bundle, and if so, how to show this?

There is a result which states that: A line bundle $L$ on a scheme $S$ is ample if and only if there exists an $n \in \mathbb{N}$ and global sections $\sigma_1, \dots, \sigma_n \in \Gamma(S, L^{\...
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67 views

Why the first Chern class of a line bundle can be non-zero

According to Chern-Weil theory Chern forms $c_i$ of the vector bundle $\xi : E \to M$ are determined by the polynomial $$ \det\left(I + \frac{\mathrm{i}t}{2\pi}F \right) = 1 + \sum^n_{i=1} c_i(\xi) ...
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73 views

Classification of line bundles over surfaces

I'm currently trying to understand the blow-up process for 4-manifolds. A step in this journey is to understand, topologically, what happens when you pluck out a 4-ball and replace it with $\mathbb{CP}...
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59 views

Line bundles correspoding to a hyperplane

Assume we have a smooth projective variety $X$ over a field and a hyperplane section $H$ on it. For each Weil divisor on $X$ you can construct a line bundle on $X$. For $H$ this line bundle which is ...
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1answer
67 views

Line bundle from $\mathcal{O}(p)\cong \mathcal{O}(q)$

Let $X$ be a compact Riemann surface. If $\mathcal{O}(p)\cong \mathcal{O}(q)$. How to see there exists a line bundle $L$ and $s_1,s_2$ two sections of $L$ such that $s_1$ vanishes only at $p$ and $s_2$...
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1answer
81 views

What is the Künneth formula for complete varieties?

I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times ...
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54 views

Global sections of square root line bundle

Let $C$ be a smooth curve in $\mathbb{P}^2$ over field $\mathbb{C}$. Suppose that I have a very ample line bundle $L$ on $C$ of even degree. Then $L$ has $2^{2g}$ square roots in $Pic\ C$. These are ...
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116 views

Picard group of a conic

Suppose $C$ is a smooth conic in $\mathbb{P^2}$, that is, $C$ is a hypersurface of degree two in the projective plane. Assume that the base field is complex numbers. By the genus degree formula, we ...
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4answers
402 views

Examples of non trivial vector bundles

Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles. Here, I want to collect such examples with justification of one or two lines saying ...
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19 views

A question about the definition of ample line bundle.

A question about the definition of ample line bundle. On a Noetherian scheme, a line bundle $L$ is said to be ample if for any coherent sheaf $F$, there exists $n_0\in\mathbb N$ such that $F\otimes ...
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89 views

Closed embedding = very ample line bundle

Let $\pi \colon X \to \mathbb{P}^n$ be a closed embedding given via an invertible sheaf $\cal L$ with global sections $s_0, \dots, s_n$. Thus ${\cal L} \cong \pi^* {\cal O}_X$. Why is ${\cal L} \...
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0answers
115 views

Bundle of Endomorphism of Line Bundle always trivial

Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact). Let $L$ be a line bundle over $B$. My question is how to see that the morphism ...
2
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1answer
70 views

Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $...